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Classification of Linear Codes by Extending Their Residuals
[chapter]
2020
Lecture Notes in Computer Science
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An approach for classification of linear codes with given parameters starting from their proper residual codes or subcodes is presented. ...
As an application, the nonexistence of binary [41, 14, 14] codes is proved. ...
We have to prove that (1) any [n, k, d] q code with the needed dual distance is equivalent to a code in the set M , and (2) the codes in M are not equivalent. (1) Let C be an [n, k, d] q code with dual ...
doi:10.1007/978-3-030-52200-1_17
fatcat:z2fsrlluanhq5e3nkjztjh3btq
Classification of the extremal formally self-dual even codes of length 30
2010
Advances in Mathematics of Communications
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Let C be a binary [n, k, d] code and let c be a codeword of weight d. Then As the residual code can be considered as a punctured code on a set of coordinates, its dual distance is at least d ⊥ . ...
Theorem 2. [8] Let C be a binary [n, k, d] code and let c be a codeword of weight Actually, we consider a residual code with respect to a codeword of minimum weight. ...
If C is a linear [n, k, d] code with dual distance d ⊥ and T is a set of t coordinates, then the dual distance of its punctured code C T is at least d ⊥ . ...
doi:10.3934/amc.2010.4.433
fatcat:vuu7kw3hozfjzaydxhf2enipm4
Results on Binary Linear Codes With Minimum Distance 8 and 10
2011
IEEE Transactions on Information Theory
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2010 pre-print arXiv:1006.0109v1 fatcat:woeynhdxkjc4xoc6zahqgulqna
All codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. ...
Primarily two algorithms considering the dual codes are used, namely extension of dual codes with a proper coordinate, and a fast algorithm for finding a maximum clique in a graph, which is modified to ...
Then Res(C, c) is an [n−w, k−1, d ′ ] code with d ′ ≥ d−w+⌈w/2⌉, and on the dual distance by Proposition 4 Suppose C is a binary [n, k, d] code with dual distance d ⊥ , c ∈ C, and the dimension of Res( ...
doi:10.1109/tit.2011.2162264
fatcat:gl4wcdplhreojbhj2v6a6lvmv4
Optimal subcodes of formally self-dual codes and their optimum distance profiles
2013
Applicable Algebra in Engineering, Communication and Computing
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In this paper we show that some optimal formally self-dual codes actually contain optimal subcodes by computing the optimum distance profiles (ODPs) of linear codes. ...
A (binary) formally self-dual code is a linear code whose weight enumerator is equal to that of its dual. Little is known about the existence of optimal subcodes of formally self-dual codes. ...
minimum distances than the minimum distances of self-dual codes. ...
doi:10.1007/s00200-013-0195-y
fatcat:zbhsql7ogba6xa7wuctp6tptwa
On the classification of quaternary optimal Hermitian LCD codes
[article]
2021
arXiv
pre-print
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We propose a method for a classification of quaternary Hermitian LCD codes having large minimum weights. ...
As an example, we give a classification of quaternary optimal Hermitian LCD codes of dimension 3. ...
[n, k, d] codes with dual distances d ⊥ ≥ 2. ...
arXiv:2011.04139v2
fatcat:ksltz7vzsnhnrahkb2sk2t7ldy
Characterization and classification of optimal LCD codes
[article]
2021
arXiv
pre-print
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2020 pre-print arXiv:1908.03294v2 fatcat:im5bu6yfpbaf7figufury4djpaOther Versions
Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. We give a characterization of LCD codes over 𝔽_q having large minimum weights for q ∈{2,3}. ...
Moreover, we give a complete classification of optimal LCD [n,k] codes over 𝔽_q for (q,k) ∈{(2,3), (2,4), (3,2),(3,3)}. ...
Theorem 4.5 (i) claims that there is a one-to-one correspondence between equivalence classes of LCD [n, k, d] codes over F q with dual distances d ⊥ ≥ 2 and equivalence classes of LCD [q · r q,n,k,d , ...
arXiv:1908.03294v3
fatcat:a3ome4kpgjhzrpn6cstjqnpd74
A simple combinatorial treatment of constructions and threshold gaps of ramp schemes
2013
Cryptography and Communications
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Finally, as an immediate consequence of these results, we provide a new explicit bound on the the minimum length of a code having a specified distance and dual distance. ...
We then generalise a construction method for ramp schemes employing error-correcting codes so that it can be applied using nonlinear (as well as linear) codes. ...
[8, 9] Suppose C is a code of length n, on an alphabet of size q, having dual distance d * . ...
doi:10.1007/s12095-013-0082-1
fatcat:5aczimg7efdjjexnoatnfs5bbi
On components of a Kerdock code and the dual of the BCH code C_1,3
[article]
2018
arXiv
pre-print
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In the paper we investigate the structure of i-components of two classes of codes: Kerdock codes and the duals of the primitive cyclic BCH code with designed distance 5 of length n=2^m-1, for odd m. ...
We prove that for any admissible length a punctured Kerdock code consists of two i-components and the dual of BCH code is a i-component for any i. ...
From (5) we see that any codeword of K ′′ n/2 is at distance d − 2 from at least one codeword of K d−2 and a codeword of K ′′ n−d−2 is at distance d − 2 from at least one codeword of K ′′ n/2 . ...
arXiv:1810.04367v1
fatcat:5ixxikmwhfhzvctnpedovzb454
An improved upper bound on self-dual codes over finite fields GF(11), GF(19), and GF(23)
[article]
2021
arXiv
pre-print
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Consequently, we improve bounds on the highest minimum distance of self-dual codes, which have not been significantly updated for almost two decades. ...
They all have new parameters with respect to self-dual codes. ...
On the other hand, self-dual codes have been the subject of much interest and are regarded as one of the most important classes of error-correcting codes. ...
arXiv:2102.08422v1
fatcat:wq6ijdqdhfem7hswtpiw6wveqa
Weight distribution of cosets of small codes with good dual properties
[article]
2017
arXiv
pre-print
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The bilateral minimum distance of a binary linear code is the maximum d such that all nonzero codewords have weights between d and n-d. ...
Let Q⊂{0,1}^n be a binary linear code whose dual has bilateral minimum distance at least d, where d is odd. ...
previous works is the requirement on the dual code to have good distance properties. ...
arXiv:1408.5681v4
fatcat:4wq2gyy53beotbqf3rp7rz3bru
Bounds for Self-Dual Codes Over Z4
2000
Finite Fields and Their Applications
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The proof of these bounds relies on a reduction to a problem of binary codes, namely that of bounding the minimum dual distance of a doubly even binary code. Academic Press ...
New bounds are given for the minimal Hamming and Lee weights of self-dual codes over 9 . ...
For instance, for n"24m#1, we have 6m#1 coe$cients and U (d/4 V#U d!1)/4 V#(d!1) constraints. For d"4m, we have only 6m! ...
doi:10.1006/ffta.1999.0258
fatcat:bof47qj7crhypnu5omwif3ayye
On the covering radius of small codes versus dual distance
[article]
2018
arXiv
pre-print
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Tietäväinen's upper and lower bounds assert that for block-length-n linear codes with dual distance d, the covering radius R is at most n/2-(1/2-o(1))√(dn) and typically at least n/2-Θ(√(dnn/d)). ...
Namely, if the dual distance d = o(n), then for sufficiently large d, almost all points can be covered with radius R≤n/2-Θ(√(dnn/d)). ...
Namely, if µ is uniformly distributed on an F 2 -linear code C ⊂ F n 2 , then µ being k-wise independent is equivalent to C having dual minimum distance at least k + 1. ...
arXiv:1707.06628v2
fatcat:zjjsy3o4kngovec6kalk3ohuvy
Optimal Binary LCD Codes
[article]
2021
arXiv
pre-print
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Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. These codes were first introduced by Massey in 1964. ...
We also present some inequalities for the largest minimum weight d_LCD(n,k) of binary LCD [n,k] codes for given length n and dimension k. ...
Acknowledgements I would like to thank Petar Boyvalenkov for focusing my attention to LCD codes and introducing me to the main literature related to this type of codes. ...
arXiv:2010.13399v2
fatcat:ua2p4ffofvefxa74soz5kypsqy
Some Results for Linear Binary Codes With Minimum Distance$5$and$6$
2005
IEEE Transactions on Information Theory
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We prove that a linear binary code with parameters [34, 24, 5] does not exist. Also, we characterize some codes with minimum distance 5. ...
Let C r = [n − d, k − 1, ≥ d/2 ] be its residual Res d (C) code and let a code with parameters [n−d+1, k −1, d ] and dual distance d ⊥ does not exist. ...
It is known that C ⊥ is an [n, n − k, d ⊥ ] code. Also, d ⊥ is called dual distance of the code. Definition 2 Let G be a generator matrix of a linear binary [n, k, d] code C. ...
doi:10.1109/tit.2005.859296
fatcat:eods7iq7dvastartmt2oju7axu
Neighborhoods of binary self-dual codes
[article]
2022
arXiv
pre-print
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Further, we show that for codelength divisible by 8 such a neighborhood consists of three self-dual codes, two of them are doubly-even and one is always singly-even. ...
In this paper, we introduce the neighborhood of binary self-dual codes. ...
If a code C is a k-dimensional subspace of F n 2 with minimum distance d, then we say that C is an (n, k, d)-code. Self-dual binary codes can be classified into Type I and Type II codes [5] . ...
arXiv:2206.05588v1
fatcat:66d5p6bqozbxpjsgdjzrvzb5pe
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